`16(costheta-cospi/8)(costheta-cos(3pi)/8)(costheta-cos(5pi)/8)(costheta-cos(7pi)/8)=lambdacos4theta,` then the value of `lambda` is _____.

Explanation

To solve the equation

\[ 16(\cos \theta - \cos \frac{\pi}{8})(\cos \theta - \cos \frac{3\pi}{8})(\cos \theta - \cos \frac{5\pi}{8})(\cos \theta - \cos \frac{7\pi}{8}) = \lambda \cos 4\theta, \]

we will follow these steps:

### Step 1: Rewrite the cosines
We can express \(\cos \frac{5\pi}{8}\) and \(\cos \frac{7\pi}{8}\) using the cosine subtraction identity:
- \(\cos \frac{5\pi}{8} = \cos(\pi - \frac{3\pi}{8}) = -\cos \frac{3\pi}{8}\)
- \(\cos \frac{7\pi}{8} = \cos(\pi - \frac{\pi}{8}) = -\cos \frac{\pi}{8}\)

Thus, we can rewrite the equation as:

\[ 16(\cos \theta - \cos \frac{\pi}{8})(\cos \theta - \cos \frac{3\pi}{8})(\cos \theta + \cos \frac{3\pi}{8})(\cos \theta + \cos \frac{\pi}{8}). \]

### Step 2: Use the product-to-sum identities
We can use the identity \( (a - b)(a + b) = a^2 - b^2 \):

\[ (\cos \theta - \cos \frac{\pi}{8})(\cos \theta + \cos \frac{\pi}{8}) = \cos^2 \theta - \cos^2 \frac{\pi}{8}, \]
\[ (\cos \theta - \cos \frac{3\pi}{8})(\cos \theta + \cos \frac{3\pi}{8}) = \cos^2 \theta - \cos^2 \frac{3\pi}{8}. \]

### Step 3: Combine the results
Now we can substitute these results back into the equation:

\[ 16 \left( \cos^2 \theta - \cos^2 \frac{\pi}{8} \right) \left( \cos^2 \theta - \cos^2 \frac{3\pi}{8} \right). \]

### Step 4: Simplify the expression
Let \(x = \cos^2 \theta\). The equation becomes:

\[ 16 (x - \cos^2 \frac{\pi}{8})(x - \cos^2 \frac{3\pi}{8}). \]

### Step 5: Expand the quadratic
Expanding this gives:

\[ 16 \left( x^2 - \left( \cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} \right)x + \cos^2 \frac{\pi}{8} \cos^2 \frac{3\pi}{8} \right). \]

### Step 6: Relate to \(\cos 4\theta\)
Using the double angle formula, we know:

\[ \cos 4\theta = 2\cos^2 2\theta - 1 = 2(2\cos^2 \theta - 1)^2 - 1. \]

### Step 7: Identify \(\lambda\)
After simplifying, we find that the coefficient of \(\cos 4\theta\) in the expanded polynomial must match \(\lambda\).

Through calculation, we find that:

\[ \lambda = 2. \]

### Final Answer
Thus, the value of \(\lambda\) is

\[ \boxed{2}. \]